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1、Non-degenerate Perturbation Theory,Problem:,cant solve exactly.,Copyright Michael D.Fayer,2007,Solutions of,complete,orthonormal set of states,with eigenvaluesand,Kronecker delta,Copyright Michael D.Fayer,2007,Expand wavefunction,and,Copyright Michael D.Fayer,2007,also have,Sum of infinite number of
2、 terms for all powers of l equals 0.,Coefficients of the individual powers of l must equal 0.,Copyright Michael D.Fayer,2007,First order correction,Want to find and.,Expand,Then,After substitution,Copyright Michael D.Fayer,2007,After substitution,Left multiply by,Copyright Michael D.Fayer,2007,We ha
3、ve,Then,Copyright Michael D.Fayer,2007,First order correction to the wavefunction,Again using the equation obtained after substituting series expansions,Left multiply by,Equals zero unless i=j.,Coefficients in expansion of ket in terms of the zeroth order kets.,Copyright Michael D.Fayer,2007,is the
4、bracket of with and.,Copyright Michael D.Fayer,2007,First order corrections,Copyright Michael D.Fayer,2007,Second Order Corrections,Using l2 coefficient,Expanding,Substituting and following same type of procedures yields,l2 coefficients have been absorbed.,Second order correction dueto first order p
5、iece of H.,Second order correction due to anadditional second order piece of H.,Copyright Michael D.Fayer,2007,Energy and Ket Corrected to First and Second Order,Copyright Michael D.Fayer,2007,Example:x3 and x4 perturbation of the Harmonic Oscillator,Vibrational potential of molecules not harmonic.A
6、pproximately harmonic near potential minimum.Expand potential in power series.,First additional terms in potential after x2 term are x3 and x4.,Copyright Michael D.Fayer,2007,perturbationc and q are expansion coefficients like l.,Copyright Michael D.Fayer,2007,In Dirac representation,First consider
7、cubic term.,Copyright Michael D.Fayer,2007,has terms with same number of raising and lowering operators.,Therefore,Copyright Michael D.Fayer,2007,Sum of the six terms,Therefore,With,Copyright Michael D.Fayer,2007,Perturbation Theory for Degenerate States,and,normalize and orthogonal,and,Degenerate,s
8、ame eigenvalue,E.,Any superposition of degenerate eigenstates is also an eigenstatewith the same eigenvalue.,Copyright Michael D.Fayer,2007,n linearly independent states with same eigenvaluesystem n-fold degenerate,Can form an infinite number of sets of.Nothing unique about any one set of n degenera
9、te eigenkets.,Can form n orthonormal,Copyright Michael D.Fayer,2007,Want approximate solution to,zeroth orderHamiltonian,perturbation,zeroth ordereigenket,zeroth orderenergy,Copyright Michael D.Fayer,2007,Here is the difficulty,perturbed ket,zeroth order ket having eigenvalue,Copyright Michael D.Fay
10、er,2007,To solve problem,Expand E and,Some superposition,but we dont know the cj.Dont know correct zeroth order function.,Copyright Michael D.Fayer,2007,To solve,substitute,Copyright Michael D.Fayer,2007,this piece becomes,Left multiplying by,Copyright Michael D.Fayer,2007,Correction to the Energies
11、,Two cases:i m(the degenerate states)and i m.,Copyright Michael D.Fayer,2007,is a system of m of equations for the cjs.,Copyright Michael D.Fayer,2007,Solve mth degree equation get the.Now have the corrections to energies.,To find the correct zeroth order eigenvectors,one for each,substitute(one at
12、a time)into system of equations.,Get system of equations for the coefficients,cjs.,There are only m 1 conditions because can multiply everything by constant.Use normalization for mth condition.,Now we have the correct zeroth order functions.,Know the.,Copyright Michael D.Fayer,2007,The solutions to
13、the mth degree equation(expanding determinant)are,Therefore,to first order,the energies of the perturbed initially degeneratestates are,Have m different(unless some still degenerate).,Copyright Michael D.Fayer,2007,Correction to wavefunctions,Again using equation found substituting the expansions intothe first order equation,Copyright Michael D.Fayer,2007,First order degenerate perturbation theory results,Correct zeroth order function.Coefficients ck determined fromsystem of equations.,Correction tozeroth order function.,Copyright Michael D.Fayer,2007,
